Page "Soundness" Paragraph 22

Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.

Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models ( up to isomorphism ) is restricted to the intended one.

The original completeness proof applies to all classical models, not some special proper subclass of intended ones.